All other inequivalent forms represent zero nontrivially (and hence their
associated orthogonal groups are non-compact [1]}),
a fact that we will exploit later.
This means, that also for \(n=4\) , both \({{\mathbb {Q}}}_p^4\)  and \({{\mathbb {R}}}^4\)  have essentially
unique groups \(SO(4)_p\)  and \(SO(4)_{{{\mathbb {R}}}}\) . On the other hand, for \(n\ge 5\) ,
it is known that for all primes \(p\) , there are no definite quadratic forms
on \({{\mathbb {Q}}}_p^n\)  [2]}, [3]}, unlike the real Euclidean case \({{\mathbb {R}}}^n\) ,
where of course the sum of the squares of the standard coordinates is the
unique definite form up to equivalence, and where we thus have an unambiguous
compact Lie group \(SO(n)_{{{\mathbb {R}}}}\) . Because of the lack of definite quadratic
forms for \(n\ge \) , we refrain from speaking of \(SO(n)_p\)  without qualification.
